The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important symmetry with applications to combinatorics, geometry, representation theory, et cetera, especially in the context of total positivity. See for instance this nice paper of Karp which contains some survey of applications of the cyclic shift: https://arxiv.org/abs/1805.06004.

**Question**: is the orbit structure of the cyclic shift acting on the *finite Grassmannian* $\mathrm{Gr}(\mathbb{F}_q;k,n)$ understood? E.g., is there a cyclic sieving phenomenon here (see https://arxiv.org/abs/1008.0790 for a survey on the CSP)?

The prototypical cyclic sieving result about $k$-subsets of $\{1,2,\ldots,n\}$ under rotation would be a ``$\mathbb{F}_1$'' version of whatever this is.